Uniform point-wise error estimates of semi-implicit compact finite difference methods for the nonlinear Schrodinger equation perturbed by wave operator

作者：Wang, Tingchun^*

来源：Journal of Mathematical Analysis and Applications, 2015, 422(1): 286-308.

DOI：10.1016/j.jmaa.2014.08.026

• NLSW
• SICFD method
• Induction argument
• Reduction of order method
• Uniform error estimate
• Pointwise error estimate

This study is devoted to analysis of semi-implicit compact finite difference (SICFD) methods for the nonlinear Schrodinger equation (NLS) perturbed by the wave operator (NLSW) with a perturbation strength described by a dimensionless parameter epsilon is an element of (0, 1]. Uniform l(infinity)-norm error bounds of the proposed SICFD schemes are built to give immediate insight on point-wise error occurring as time increases, and the explicit dependence of the mesh size and time step on the parameter a is also figured out. In the small is an element of regime, highly oscillations arise in time with O(epsilon(2))-wavelength. This highly oscillatory nature in time as well as the difficulty raised by the compact FD discretization make establishing the l(infinity)-norm error bounds uniformly in a of the SICFD methods for NLSW to be a very interesting and challenging issue. The uniform l(infinity)-norm error bounds in a are proved to be of O(h(4) + tau) and O(h(4) + tau(2/3)) with time step tau and mesh size h for well-prepared and ill-prepared initial data. Finally, numerical results are reported to verify the error estimates and show the sharpness of the convergence rates in the respectively parameter regimes.

• 出版日期2015-2-1
• 单位南京信息工程大学